1. Stat 31, Section 1, Last Time Linear transformations
Standardization (subt. mean / div. be SD)
5 Number Summary & Outlier Rule
Modelling distributions density
Normal distributions
Density
Interpretation
Computation

2. Computation of Normal Areas
Classical Approach: Tables
See inside covers of text
Summarizes area computations
Because can’t use calculus
Constructed by “computers”
(a job description in the early 1900’s!)

3. Computation of Normal Areas
EXCEL Computation:
works in terms of “lower areas”
E.g. for
Area < 1.3

4. Computation of Normal Areas
Interactive Version (used for above pic)
From Webster West’s Website:

5. Computation of Normal Areas
EXCEL Computation:
(of above e.g.)
Enter parameters
x is “cutoff point”
Return is Area below x

6. Computation of Normal Areas
Computation of areas over intervals:
(use subtraction) =

7. Computation of Normal Areas
Computation of areas over intervals:
(use subtraction for EXCEL too)
E.g. Use Excel to check % Rule

8. Normal Area HW HW: 1.89 1.92 (Hint: the % above 500 =
100% - % below 500)
1.97
(50%, 9.18%, 0.38%, 40.82%)
Caution: Don’t just “twiddle EXCEL until answer appears”. Understand it!!!

9. Inverse of Area Function
Inverse of Frequencies: “Quantiles”
Idea: Given area, find “cutoff” x
I.e. for
Area = 80%
This x
is the “quantile”

10. Inverse of Area Function
EXCEL Computation of Quantiles:
Use NORMINV
Continue Class Example:
“Probability” is “Area”
Enter mean and SD parameters

11. Inverse Area Example
When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz.
The machine is “out of control” when it overfills. Choose an “alarm level”, which will give only 1 % false alarms.
Want: cutoff, x, so that Area above = 1%
Note: Area below = 100% - Area above = 99%

12. Inverse Area HW
1.105
(-0.675, 0.675, 89.9, 110.1, 1.35, 0.7%)
1.107

13. Normal Diagnostic When is the Normal Model “good”?
Useful Graphical Device:
Q-Q plot = Normal Quantile Plot
Won’t devote class time:
Useful info in text about this

14. Variable Relationships
Chapter 2 in Text
Idea: Look beyond single quantities, to how quantities relate to each other.
E.g. How do HW scores “relate”
to Exam scores?
Section 2.1: Useful graphical device:
Scatterplot

15. Recall Scatterplot E.g.
Toy Example:
(1,2)
(3,1)
(-1,0)
(2,-1)

16. Scatterplot E.g. Data from related Intro. Stat. Class (actual scores)
How does HW score predict Final Exam?
= HW, = Final Exam
In top half of HW scores:
Better HW  Better Final
For lower HW:
Final is much more “random”

17. Scatterplots Common Terminology: When thinking about “X causes Y”,
Call X the “Explanatory Var.” or “Indep. Var.”
Call Y the “Response Var.” or “Dep. Var.”
(think of “Y as function of X”)
(although not always sensible)

18. Scatterplots Note: Sometimes think about causation,
Other times: “Explore Relationship”
HW:

19. Class Scores Scatterplots
How does HW predict Midterm 1?
= HW, = MT1
Still better HW  better Exam
But for each HW, wider range of MT1 scores
I.e. HW doesn’t predict MT1 as well as Final
“Outliers” in scatterplot may not be outliers in either individual variable
e.g. HW = 72, MT1 = 94
(bad HW, but good MT1?, fluke???)

20. Class Scores Scatterplots
How does MT1 predict MT2?
= MT1, = MT2
Idea: less “causation”, more “exploration
Still higher MT1 associated with higher MT2
For each MT1, wider range of MT2
i.e. “not good predictor”
Interesting Outliers:
MT1 = 100, MT2 = 56 (oops!)
MT1 = 23, MT2 = 74 (woke up!)

21. Important Aspects of Relations
Form of Relationship
Direction of Relationship
Strength of Relationship

22. I. Form of Relationship Linear: Data approximately follow a line
Previous Class Scores Example
Final vs. High values of HW is “best”
Nonlinear: Data follows different pattern
Nice Example: Bralower’s Fossil Data

23. Bralower’s Fossil Data
From T. Bralower, formerly of Geological Sci.
Studies Global Climate, millions of years ago:
Ratios of Isotopes of Strontium
Reflects Ice Ages, via Sea Level
(50 meter difference!)
As function of time
Clearly nonlinear relationship

24. II. Direction of Relationship
Positive Association
X bigger  Y bigger
Negative Association
X bigger  Y smaller
E.g. X = alcohol consumption, Y = Driving Ability
Clear negative association

25. III. Strength of Relationship
Idea: How close are points to lying on a line?
Revisit Class Scores Example:
Final Exam is “closely related to HW”
Midterm 1 less closely related to HW
Midterm 2 even related to Midterm 1

26. Linear Relationship HW
2.3, 2.5, 2.7, 2.9